With the development of technological and experimental techniques , crystals with all kinds of dimension and shape have been made by experiment, which has brought out a large market for the apply of new materials. So there is continuing interests in polaron. LeeLowPines calculated the groundstate energy of polaron by a variational technique. Huybrechts calculated the energy of the groundstate and the first internal excited state of the optical polaron for different values of the electronphonon coupling. N.Tokuda studied the dependence of groundstate energy, effective mass and the mean number of polaron on the coupling constant by using the unitary transformation and the method of a lagrange multiplier. Larsen got the groundstate energy of the twodimension magnetoplaron by using forthorder perturbationtheoretic method. Wei et al. investigated the cyclotronresonance mass and frequency of the interface polaron by using Feyman path integration method. Recently, Catandella et al. studied the properties of polaron in onedimension Holstein molecular crystals by using a new variational method, which chosen the linear overlap of Bloch wave of large polaron and small polaron as a trial wavefunction, obtained the variational regularities of the groundstate energy, mass and the mean number of polaron with coupling constant. We also did some works on polaron by using Huybrechts' method. But all works were based on only one considering either magnetic field or coupling strength,respectively. This paper is going to illustrate the common influence of magnetic field and coupling strength on the properties of polaron.The variational relations of the groundstate energy, selftrapping energy and Landau energy with coupling constant and cyclotronresonance frequency are derived by using linearcombination operator method. Numerical calculation indicates that the vibration frequency λ firstly falls, arriving to a minimum value, late increases with increasing coupling constant α; λ monotonously rises with increasing cyclotronresonance frequency ωc;the groundstate energy E0 decrease with increasing α and monotonous rises with increasing ωc;the selftrapping energy E0tr increases with increasing α and ωc;Landau energy E0L firstly increases to a maximum and then fall with increasing α;E0L firstly rises to maximum also and then decreases with increasing ωc.
With the development of technological and experimental techniques , crystals with all kinds of dimension and shape have been made by experiment, which has brought out a large market for the apply of new materials. So there is continuing interests in polaron. LeeLowPines calculated the groundstate energy of polaron by a variational technique. Huybrechts calculated the energy of the groundstate and the first internal excited state of the optical polaron for different values of the electronphonon coupling. N.Tokuda studied the dependence of groundstate energy, effective mass and the mean number of polaron on the coupling constant by using the unitary transformation and the method of a lagrange multiplier. Larsen got the groundstate energy of the twodimension magnetoplaron by using forthorder perturbationtheoretic method. Wei et al. investigated the cyclotronresonance mass and frequency of the interface polaron by using Feyman path integration method. Recently, Catandella et al. studied the properties of polaron in onedimension Holstein molecular crystals by using a new variational method, which chosen the linear overlap of Bloch wave of large polaron and small polaron as a trial wavefunction, obtained the variational regularities of the groundstate energy, mass and the mean number of polaron with coupling constant. We also did some works on polaron by using Huybrechts' method. But all works were based on only one considering either magnetic field or coupling strength,respectively. This paper is going to illustrate the common influence of magnetic field and coupling strength on the properties of polaron.The variational relations of the groundstate energy, selftrapping energy and Landau energy with coupling constant and cyclotronresonance frequency are derived by using linearcombination operator method. Numerical calculation indicates that the vibration frequency λ firstly falls, arriving to a minimum value, late increases with increasing coupling constant α; λ monotonously rises with increasing cyclotronresonance frequency ωc;the groundstate energy E0 decrease with increasing α and monotonous rises with increasing ωc;the selftrapping energy E0tr increases with increasing α and ωc;Landau energy E0L firstly increases to a maximum and then fall with increasing α;E0L firstly rises to maximum also and then decreases with increasing ωc.
The varied relation of effective mass and mean number of phonon of polaron with coupling strength and magnetic field was derived by using the linear combination operator and unitary transformation methods.The numerical calculation indicates that the vibration frequency λ firstly falls and late increase with increasing coupling constant α,λ monotonous increases with increasing magnetic field strength;effective mass of polaron monotonous increases with increasing coupling strength and occurs resonance with varing magnetic field strength.In resonance point,because coupling between electron and crystal vibration is highest strength,effective mass of polaron achieves maximum.The numerical calculation also shows that mean number of phonon of polaron monotonous increases with increasing coupling strength, but decreases with increasing magnetic field strength when magnetic field strength is larger than magnetic strength of resonance point,since the farther distance from resonance point is,the weaker the interaction between electron and phonons is,conclusion is contrary when magnetic field strength is smaller than magnetic strength of resonance point.
With the improvement of experimental techniques and the replacement of equipment, composition and application of solid material have been gained a series new development. So there is continuing interests in polaron. N.Tokuda discussed that the variational law of the groundstate energy, the excitedstate energies with coupling constant; Xiao Jinglin, et al. investigated that the properties of polaron in various coupling strength; Xiao Wei, et al. studied that the affact of temperature on polaron; We discussed that the properties of acoustic deformation potential polaron. This paper was going to illustrate the variational law of the effective mass of polaron with temperature and coupling strength.In this paper, the variational relation of the effective mass of polaron in crystals with temperature and coupling strength was derived by using linearcombination operator and unitary transformation method. The results indicated that the effective mass increases with increasing coupling strength when temperature is constant, resulting in the interaction of electronphonon strengthens with increasing coupling strength. The results also illustrate that the effective mass varies with variation of temperature when coupling strength is constant, the stronger coupling strength is, the more obvious the affect of temperature on effective mass. At limit of weakcoupling,the effective mass of slowmoving polaron has nothing to do with temperature because a electron is approximate a freeelectron at limit of weakcoupling and it's motion is not affected by crystals vibration.In sum, the effective mass of polaron increases with increasing coupling strength when temperature is constant, and affect of temperature on effective mass also increases with increasing coupling strength.
As ones know,the polaron is a quasiparticle formed by an electron and phonons.Due to Coulomb interaction,the electron will make the lattice deformation take place,meanwhile,the deformation potential(induced) by the lattice deformation will exert on the electron,the electron-phonon interaction influence doubtless on the state of motion of an electron,for example,the energy band and effective mass of an electron could change by the interaction,the electron can also be trapped by the deformable lattice,therefore,the research on the(polaron) is very important in understanding optical and transport properties of a solid. Studying electron-phonon interaction has been a main subject in condensed matter physics;in experiments,the influences of the temperature on the forbidden-band width of semiconductors have been observed very early,with the increase of the temperature,the forbidden-band width of most semiconductors will narrow,but a few will widen,these situations are closely relevant to the electron-phonon interaction.In particular,(recently),a large amount of experimental results,ranging from infrared spectroscopy to transport properties(involving) the colossal magnetoresistance and high-T_c superconductivity,has pointed out the importance of the electron-phonon interaction,which has renewed the interest in studying the models of the electron-phonon(coupled) system,especially,the Holstein polaron,for its relative simplicity,is the most considered model for the interaction of a single tight-binding electron coupled to phonons. In the past several decades,the Holstein poloron has been studied in detail in one-dimensional lattices,various theoretical methods,such as variational approaches,the Global-Local variational method,the(coherent)-state expansion method,have been developed to study features of the Holstein polaron including ground-state energy,polaron energy band,effective mass and so on.However,so far,the work considering the influences of the temperature on features of the Holstein polaron is little still. The main aim of this paper is to investigate the influences of the temperature on the energy-band width and the effective mass of the Holstein polaron by using the canonical transformation method.It is shown that,at certain coupling strength,the energy-band width of the Holstein polaron will narrow and the effective mass will augment with the increase of the temperature,in particular,when electron-phonon coupling strength is big enough,the energy-band width will become zero rapidly in small region of temperature,in fact,the situation is rapid transition of the polaron from the energy-band state to the self-trapping local state,this is an(interesting) result obtained in the paper.It is obvious that these features of the Holstein polaron are very useful in explaining optical and transport properties of a solid.
The polaron band theory of Lee, Low and Pines is extended beyond the effective mass approximation to a general form of the electron–phonon interaction within two different self-consistent schemes. In the first one a variational Bloch state depending on the polaron crystalline momentum is used to study polaron features taking into account umklapp processes. In particular, the self-energy and the effective mass at the Γ point of the Brillouin zone are calculated as functions of the electron–phonon coupling constant α. It is found that a sudden transition of these quantities from a large band to a narrow band polaron behaviour occurs at a critical value αc. Such value increases indefinitely with the bare electron band width; moreover the transition becomes sharper. The theory recovers, as particular cases, a) the Lee-Low-Pines regime valid for small α and large electron band width Δ˜(Δ˜ ≫ ħω and b) the Holstein limit for large α and small band width (Δ˜ ≫ ħω). This occurs as result of the competing role of α and Δ˜. In the second scheme, the electron–phonon interaction is considered only in a single elementary cell and the self-trapped polaron state is calculated self-consistently. It is shown that, if only one electronic band is taken into account, such state becomes a Δ function with infinitely negative energy when α is larger than a critical value depending on the electron band width. To avoid this fact, electronic band states of higher energy must be considered. Finally, through the tight-binding method, the polaron band is set up. It is shown that at the Γ point of the Brillouin zone, the contribution of the higher electronic bands prevents the self-energy and the polaron mass to have a sharp increase from the large to the narrow band regime.
A variational technique is developed to investigate the polaronic features of the Holstein molecular-crystal model. It is based on a linear superposition of Bloch states that describe large and small polaron wave functions. It is shown that this method provides a very good description of the regime characterized by intermediate values of the electron-phonon coupling constant (the so-called intermediate polaron) for any value of the adiabatic parameter ${\ensuremath{\omega}}_{0}/t.$ The polaron ground state energy in one and two dimensions is calculated and successfully compared with the best estimates available providing a clear physical interpretation of the intermediate polaron. The band structure, the spectral weight of the ground state and the lattice displacement associated to the polaron are also calculated and discussed. The method has the advantage of requiring very little computational effort.
It is well known that two qualitatively different pictures arise in the description of the motion of an electron in an ionic crystal. In the first case, when the electron interacts weakly with the ionic lattice of the crystal, its motion is the same as the motion of a free band electron whose energy is shifted downward with respect to the bottom of the conduction band, and the effective mass is replaced by the renormalized mass (weak-coupling polaron). In the other limiting case, when the electron interacts strongly with the ionic crystal, an entire series of different self-consistent states of the electron and lattice, each of which has its own effective mass and radius (strong-coupling polaron), arises. Although the motion of an electron in an ionic crystal is described in both limiting cases with the help of the Pekar-Frohlich Hamiltonian, each case requires its own special methods of analysis. Thus, for example, in the strong-coupling theory a special form of adiabatic perturbation theory has been developed in which the trans-lational degeneracy is removed before the perturbation expansion is made. It has been possible to relate both descriptions within the framework of Feynman's varia-tional approach, in which a smooth upper bound was obtained for the energy of the ground state of the polaron for all values of the electron-phonon coupling constant a (see Fig. 1). We call attention in this connection to Ref. 6, where the thermodynamic properties are calculated with the help of a special method for performing the calculations and for finding averages of T-products. The approach developed there is more convenient for performing specific calculations than is the use of path integrals. It seems to us that it has definite advantages in the construction of general proofs concerning the choice of the approximating Hamiltonians (this is illustrated in Ref. 6 for the example of the linear Bogolyubov model), in the investigation of the characteristics of the behavior of a polaron in external fields, etc.